硬核模型树单域中MCMC和环BP的收敛性
Convergence of MCMC and Loopy BP in the Tree Uniqueness Region for the Hard-Core Model
摘要：
我们研究了定义在输入图的独立集上的硬核模型，其中独立集由参数$λ>0$加权。对于常数$Δ$，Weitz（2006）的先前工作建立了当$λ< λ_c(Δ)$时最大度$Δ$的图的配分函数的FPTAS。阈值$λ_c(Δ)$是无限$Δ$ -正则树上唯一性/非唯一性相变的临界点。Sly（2010）表明，当$λ>λ_c(Δ)$时，除非NP=RP，否则不存在FPRAS。Weitz算法在$\log(Δ)$中的运行时间是指数的。在这里，我们提出了一个FPRAS的配分函数，其运行时间是$O^*(n^2)$。我们分析了简单的单站点Glauber动态采样从相关的吉布斯分布。证明了存在一个常数$Δ_0$，使得对于所有具有最大度$Δ\geqΔ_0$和围长$\geq 7$的图，当$λ<λ_c(Δ)$时，Glauber动力学的混合时间为$O(n\log(n))$.我们的工作是对Weitz的工作的补充，Weitz的工作适用于常数$Δ$，而我们的工作适用于所有$Δ\geq Δ_0$。我们利用循环BP（信念传播），广泛使用的推理算法。我们工作的一个新的方面是使用BP算子的主特征向量来设计一个距离函数，该距离函数对于表现为BP不动点的状态对在期望中收缩。我们还证明了Glauber动力学的局部行为像循环BP。作为一个副产品，我们得到的Glauber动力学收敛，经过一个短的老化期，接近BP固定点，这意味着循环BP的固定点是一个密切的近似吉布斯分布。使用这些连接，我们建立循环BP快速收敛到吉布斯分布时，周长$\geq 6$和$λ<λ_c(Δ)$。
英文摘要：
We study the hard-core model defined on independent sets of an input graph where the independent sets are weighted by a parameter $λ>0$. For constant $Δ$, previous work of Weitz (2006) established an FPTAS for the partition function for graphs of maximum degree $Δ$ when $λ< λ_c(Δ)$. The threshold $λ_c(Δ)$ is the critical point for the phase transition for uniqueness/non-uniqueness on the infinite $Δ$-regular trees. Sly (2010) showed that there is no FPRAS, unless NP=RP, when $λ>λ_c(Δ)$. The running time of Weitz's algorithm is exponential in $\log(Δ)$. Here we present an FPRAS for the partition function whose running time is $O^*(n^2)$. We analyze the simple single-site Glauber dynamics for sampling from the associated Gibbs distribution. We prove there exists a constant $Δ_0$ such that for all graphs with maximum degree $Δ\geqΔ_0$ and girth $\geq 7$, the mixing time of the Glauber dynamics is $O(n\log(n))$ when $λ<λ_c(Δ)$. Our work complements that of Weitz which applies for constant $Δ$ whereas our work applies for all $Δ\geq Δ_0$. We utilize loopy BP (belief propagation), a widely-used inference algorithm. A novel aspect of our work is using the principal eigenvector for the BP operator to design a distance function which contracts in expectation for pairs of states that behave like the BP fixed point. We also prove that the Glauber dynamics behaves locally like loopy BP. As a byproduct we obtain that the Glauber dynamics converges, after a short burn-in period, close to the BP fixed point, and this implies that the fixed point of loopy BP is a close approximation to the Gibbs distribution. Using these connections we establish that loopy BP quickly converges to the Gibbs distribution when the girth $\geq 6$ and $λ<λ_c(Δ)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
